Derive an equation for electric potential V(z) on z axis

AI Thread Summary
The discussion revolves around deriving the electric potential V(z) on the z-axis due to four negative charges, using the equation V=kQ/r. The user has calculated the distances r12 and r34 for the charges but struggles to express V(z) in the desired form, particularly how to achieve a negative result for V(z) despite both distance and z being positive. They have derived an equation V(z)=-Q/8*pi*Enod(sqrt([d^2+(2Z-d)^2])) but seek clarification on how to adjust it to match the expected format. The user is looking for further insights into their calculations and assistance with additional parts of the homework. Overall, the thread highlights challenges in understanding electric potential equations in the context of multiple point charges.
jackMybrain@ru
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Homework Statement


Hi, I am stuck with this homework. I have been asked to make an equation out of a diagram for V(z) using V=kQ/r equation, where z is a positive axis centred with four negative charges. Here is the diagram.
9f12337b48.png


2. Relevant equation
E=F/q
F of point charge: F=kQq/r^2
E=kQ/r^2
V=kQq/r ~U=V*q & KE=0.5m*velocity^2

r=distant of the point from the point charge q
k=1/4*pi*[E][/0]

The Attempt at a Solution


I have already completed the (a) and (b) parts. However, for part (b) I could only make V(z)=Q/(4*pi*Enod)*r where r = 0.5*sqrt([d^2+(2Z-d)^2]) from the r12. I found r by taking all point charges distant together by saying r12 of the first 2 charges are equal and r34 of the 3rd and 4th is same. Therefore, Vtotal=KQ*[(2/r12)+(2/r34)]. But how do I make it as V(z)=-Q/pi*Enod... because currently I am getting V(z)=-Q/8*pi*Enod(sqrt([d^2+(2Z-d)^2])).
Am I doing something wrong. Thanks for any insight or solution for the other parts would be very helpful.
And how do I do Q 1. (c),(d)& (f)??
 
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jackMybrain@ru said:

Homework Statement


Hi, I am stuck with this homework. I have been asked to make an equation out of a diagram for V(z) using V=kQ/r equation, where z point start in the center with four negative charges. Here is the diagram for the aid.
9f12337b48.png


2. Relevant equation
E=F/q
Force of point charges by Coulumbs law: F=kQq/r^2
E=kQ/r^2
V=kQq/r ~U=V*q & KE=0.5*mass*velocity^2

r=distant of the point from the point charge q
k=1/4*pi*Enod

The Attempt at a Solution


I have already completed the (a) and (b) parts.
However, for part (b) I don't understand how to make a negative result for V(z)??
As normal equation goes, I have derived only V(z)=Q/(4*pi*Enod)*r where r = 0.5*sqrt([d^2+(2Z-d)^2]) (from the r12).
When I put everything together my found equation: V(z)=-Q/8*pi*Enod(sqrt([d^2+(2Z-d)^2])).
But how do I make it as V(z)=-Q/pi*Enod.(...) because both distance and z axis are positive.

**I found my d or r by taking all point charges distant from z together. For instance accumulating r12 of q1 and 12 are equal apart from z on z axis and r34 of the 3rd and 4th are higher than them vice versa. Therefore, (r) total=(2/r12)+(2/r34). So, Vtotal=KQ/[(2/r12)+(2/r34)].

Am I doing something wrong. I am only trying to understand physics. Thanks for any insight or solution for the other parts would be very helpful.
And how do I do Q 1. (c),(d)& (f)??

This is an easier version of my question. Please, help! I don't know why nobody is familiar with V(z) equation or how to solve them. Because there are not many references or videos to be found by online searching. Thanks again..
 
Show your derivation for r12 and r34 in detail.
 
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